On q-multiplicative functions taking a fixed value on set of primes

The following assertion is proved. Let gbe a q-multiplicative function for which |g(n)| = 1 for every integer, and g(p) = constantfor every large prime p. Then there is some integer kin [1; c] such that g ^k(nq) = 1 for every n∈ N. Here cis a suitable absolute constant.     

Finite groups with three conjugacy class sizes of some elements

Let G be a finite group. We prove as follows: Let G be a p-solvable group for a fixed prime p. If the conjugacy class sizes of all elements of primary and biprimary orders of G are {1,p ^a , n} with a and n two positive integers and (p,n)?=?1, then G is p-nilpotent or G has abelian Sylow p-subgroups.     

Abelian group matrices over the p-adic and rational integers

Let G be a finite abelian group of order n. Let Z and Q denote the rational integers and rationals, respectively. A group matrix for G over Z (or Q) is an n-square matrix of the form Σ_{g ∈ G}a_gP(g), where a_g ∈ Z (or Q) and P is the regular representation of G so that P(g) is an n-square permutation matrix and P(gh) = P(g)P(h) for all g, h ∈ G. It is known that if M is an arbitrary positive definite unimodular matrix over Z then there exists a matrix A over Q such that M = A^τA, where τ denotes transposition. This paper proves that the exact analogue of this theorem holds if one demands that M and A be group matrices for G over Z and Q, respectively. Furthermore, if M is a group matrix for G over the p-adic integers then necessary and sufficient conditions are given for the existence of a group matrix A for G over the p-adic numbers such that M = A^τA.     

Most small $${\varvec{}}$$-grous have an automorhism of order 2

Let f(p, n) be the number of pairwise nonisomorphic p-groups of order \(p^n\), and let g(p, n) be the number of groups of order \(p^n\) whose automorphism group is a p-group. We prove that the limit, as p grows to infinity, of the ratio g(p, n) / f(p, n) equals 1/3 for \(n=6,7\).     

A limit theorem for order preserving nonexpansive operators inL 1

We prove the following theorem:Let T be an order preserving nonexpansive operator on L ₁ (μ) (or L ₁ ⁺ ) of a σ-finite measure, which also decreases theL _∞-norm, and let S=tI+(1?t)T for 0<t<1. Then for everyf ∈ L_p (1<p<∞),the sequence S ⁿf converges weakly in L_p. (The assumptions do not imply thatT is nonexpansive inL _p for anyp>1, even ifμ is finite.) For the proof we show that ∥S ⁿ⁺¹ f?S ⁿf∥ _p → 0 for everyf ∈L _p, 1<p<∞, and apply toS the following theorem:Let T be order preserving and nonexpansive in L ₁ ⁺ , and assume that T decreases theL _∞-norm. Then forg ∈L _p (1<p<∞) Tⁿg is weakly almost convergent. If forf ∈ L_p we have T ⁿ⁺¹ f?T ⁿ f → 0weakly, then T ⁿf converges weakly in L_p (1<p<∞).     

A note on circulant transition matrices in Markov chains

Let g and n be positive integers and gcd(g,n)=1. Let C=(c_ij) be a g-circulant transition matrix of order n of Markov chain. We are interested in studying and lim_k→∞C^k.     

Approximations by convolutions and antiderivatives

Let g be a given function in L ¹ = L ¹(0, 1), and let B be one of the spaces L ^p (0, 1), 1 ≤ p < ∞, or C ₀[0, 1]. We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on g, we prove the equivalence in B of the systems f _n * g and I f _n , where f _n ∈ L ¹, n ∈ ?, and I f = f * 1 is the antiderivative of f. As a consequence, we obtain criteria for the completeness and basis property in B of subsystems of antiderivatives of g.     

On the class numbers of cyclotomic fields

Let g_n denote the first factor of the class number of the nth cyclotomic field. It is proved that if n runs through a sequence of prime powers p^r tending to infinity, then

$\text{log}\text{g}{}_{\mathrm{n}}\text{～}\text{1}\text{4}\text{[1 ? (}\text{1}\text{p}\text{)]n}\text{log}\text{n}$

.     

Conjugacy class sizes and solvability of finite groups

Let G be a finite group and let G* be the set of elements of primary, biprimary and triprimary orders of G. We show that suppose that the conjugacy class sizes of G* are exactly {1, p ^a , n, p ^a n} with (p, n)?=?1 and a??? 0, then G is solvable.     

Eigenvalue estimate for the weighted p-Laplacian

Let M be an n-dimensional closed manifold with metric g, dμ = e ^h(x) dV(x) be the weighted measure and ? _{μ, p} be the weighted p-Laplacian. In this article, we get the lower bound estimate of the first nonzero eigenvalue for the weighted p-Laplacian when the m-dimensional Bakry-émery curvature has a positive lower bound.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Weyl-Heisenberg Riesz Bases Generated by Two Intervals

Let {e ^2πimx g(x?n)} _m,n∈? be a Weyl-Heisenberg system and let g be the characteristic function of a union of two separated intervals. We systematically study the window function g such that the Weyl-Heisenberg system consists of a Riesz basis of L ²(?).     

Finite groups with the set of the number of subgroups of possible order containing exactly two elements

Let G be a finite group, and n(G) be the set of the number of subgroups of possible order of G. We investigate the structure of G satisfying that n(G)?=?{1, m} for any positive integer m?>?1. At first, we prove that the nilpotent length of G is less than 2. Secondly, we investigate nilpotent groups with m?=?p?+?1 or p ²?+?p?+?1 (p is a prime), and we get the classification of such kinds of groups. At last, we investigate non-nilpotent groups with m?=?p?+?1 and get the classification of the groups under consideration.     

Finiteness of Some Sharply Doubly Transitive Groups

Let G be a doubly transitive permutation group such that its point stabilizer is a 2-group and its two-point stabilizer is trivial. It is proved that G is finite and isomorphic to a Frobenius group of order 3²· 2³ or p· 2ⁿ, where p=2ⁿ+1 is a Fermat prime.     

Uniqueness and non-uniqueness of metrics with prescribed scalar and mean curvature on compact manifolds with boundary

Let (Mⁿ,g) be a compact manifold with boundary with n?2. In this paper we discuss uniqueness and non-uniqueness of metrics in the conformal class of g having the same scalar curvature and the mean curvature of the boundary of M.     

Characterization of convex domains in ℂ<Superscript>2</Superscript> with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ?ⁿ. Suppose there exists {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ?² with C ² boundary. Suppose that there is a sequence {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

Chen’s Conjecture and Its Generalization

Let l1, l2, ..., lg be even integers and x be a sufficiently large number. In this paper, the authors prove that the number of positive odd integers k ≤ x such that （k ＋l1）^2, （k ＋l2）^2, ..., （k ＋lg）^2 can not be expressed as 2^n＋p^α is at least c（g）x, where p is an odd prime and the constant c（g） depends only on g.     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

Polynomials and primitive roots in finite fields

The primitive elements of a finite field are those elements of the field that generate the multiplicative group of k. If f(x) is a polynomial over k of small degree compared to the size of k, then f(x) represents at least one primitive element of k. Also f(x) represents an lth power at a primitive element of k, if l is also small. As a consequence of this, the following results holds.Theorem. Let g(x) be a square-free polynomial with integer coefficients. For all but finitely many prime numbers p, there is an integer a such that g(a) is equivalent to a primitive element modulo p.Theorem. Let l be a fixed prime number and f(x) be a square-free polynomial with integer coefficients with a non-zero constant term. For all but finitely many primes p, there exist integers a and b such that a is a primitive element and f(a) ≡ b¹ modulo p.     

Widths and optimal sampling in spaces of analytic functions

Let Ω be a finitely-connected planar domain and μ be a positive measure with compact supportE in Ω. LetA _p be the unit ball of the Hardy spaceH ^p. The main result of this paper is that Kolmogorov, Gelfand, and linearn-widths ofA _p inL ^q are comparable in size to each other and to the sampling error ifq≤p. Moreover, ifp=q=2 andE is small enough, then all these quantities are equal.